Fuzzy Linear Programming

  • Masatoshi Sakawa
  • Hitoshi Yano
  • Ichiro Nishizaki
Chapter
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 203)

Abstract

In 1976, Zimmermann first introduced fuzzy set theory into linear programming problems. He considered linear programming problems with a fuzzy goal and fuzzy constraints. Following the fuzzy decision proposed by Bellman and Zadeh (1970) together with linear membership functions, he proved that there exists an equivalent linear programming problem.

Keywords

Expense 

References

  1. Bellman, R. E., & Zadeh, L. A. (1970). Decision making in a fuzzy environment. Management Science, 17, 141–164.CrossRefGoogle Scholar
  2. Carlsson, C., & Fullér, R. (2002). Fuzzy reasoning in decision making and optimization. Heidelberg: Physica-Verlag.CrossRefGoogle Scholar
  3. Delgado, M., Kacprzyk, J., Verdegay, J. L., & Vila, M. A. (Eds.) (1994). Fuzzy optimization: Recent advances. Heidelberg: Physica-Verlag.Google Scholar
  4. Dubois, D., & Prade, H. (1978). Operations on fuzzy numbers. International Journal of Systems Science, 9, 613–626.CrossRefGoogle Scholar
  5. Dubois, D., & Prade, H. (1980). Fuzzy sets and systems: Theory and application. New York: Academic.Google Scholar
  6. Ehrgott, M., & Gandibleux, X. (Eds.) (2002). Multiple criteria optimization –state of the art annotated bibliographic surveys. Boston: Kluwer.Google Scholar
  7. Fiacco, A. V. (1983). Introduction to sensitivity and stability analysis in nonlinear programming. New York: Academic.Google Scholar
  8. Haimes, Y. Y., & Chankong V. (1979). Kuhn–Tucker multipliers as trade-offs in multiobjective decision-making analysis. Automatica, 15, 59–72.CrossRefGoogle Scholar
  9. Kacprzyk, J., & Orlovski S. A. (Eds.) (1987). Optimization models using fuzzy sets and possibility theory. Dordrecht: D. Reidel Publishing Company.Google Scholar
  10. Kahraman, C. (Ed.) (2008). Fuzzy multi-criteria decision making: Theory and applications with recent developments. New York: Springer.Google Scholar
  11. Lai, Y. J., & Hwang, C. L. (1992). Fuzzy mathematical programming. Berlin: Springer.CrossRefGoogle Scholar
  12. Luenberger, D. G. (1973). Linear and nonlinear programming (2nd ed. (1984), 3rd ed. (2008)). California: Addison-Wesley.Google Scholar
  13. Sakawa, M. (1993). Fuzzy sets and interactive multiobjective optimization. New York: Plenum Press.CrossRefGoogle Scholar
  14. Sakawa, M. (2000). Large scale interactive fuzzy multiobjective programming. Heidelberg: Physica-Verlag.CrossRefGoogle Scholar
  15. Sakawa, M. (2001). Genetic algorithms and fuzzy multiobjective optimization. Boston: Kluwer.Google Scholar
  16. Sakawa, M., & Yano, H. (1985a). Interactive fuzzy decision-making for multi-objective nonlinear programming using reference membership intervals. International Journal of Man-Machine Studies, 23, 407–421.CrossRefGoogle Scholar
  17. Sakawa, M., & Yano, H. (1985b). Interactive decision making for multiobjective linear fractional programming problems with fuzzy parameters. Cybernetics and Systems: An International Journal, 16, 377–394.CrossRefGoogle Scholar
  18. Sakawa, M., & Yano, H. (1985c). Interactive fuzzy satisficing method using augmented minimax problems and its application to environmental systems. IEEE Transactions on Systems, Man and Cybernetics, SMC-15, 720–729.CrossRefGoogle Scholar
  19. Sakawa, M., & Yano, H. (1986c). Interactive fuzzy decision making for multiobjective nonlinear programming using augmented minimax problems. Fuzzy Sets and Systems, 20, 31–43.CrossRefGoogle Scholar
  20. Sakawa, M., & Yano, H. (1986d). Interactive decision making for multiobjective linear programming problems with fuzzy parameters. In G. Fandel, M. Grauer, A. Kurzhanski, & A. P. Wierzbicki (Eds.), Large-scale modeling and interactive decision analysis (pp. 88–96). New York: Springer.CrossRefGoogle Scholar
  21. Sakawa, M., & Yano, H. (1987). An interactive satisficing method for multiobjective nonlinear programming problems with fuzzy parameters. In M. Kacprzyk, & S. A. Orlovski (Eds.), Optimization models using fuzzy sets and possibility theory (pp. 258–271). Dordrecht: D. Reidel Publishing Company.CrossRefGoogle Scholar
  22. Sakawa, M., & Yano, H. (1988). An interactive fuzzy satisficing method for multiobjective linear fractional programming problems. Fuzzy Sets and Systems, 28, 129–144.CrossRefGoogle Scholar
  23. Sakawa, M., & Yano, H. (1989). Interactive decision making for multiobjective nonlinear programming problems with fuzzy parameters. Fuzzy Sets and Systems, 29, 315–326.CrossRefGoogle Scholar
  24. Sakawa, M., & Yano, H. (1990). An interactive fuzzy satisficing method for generalized multiobjective linear programming problems with fuzzy parameters. Fuzzy Sets and Systems, 35, 125–142.CrossRefGoogle Scholar
  25. Sakawa, M., Yano, H., & Yumine, T. (1987). An interactive fuzzy satisficing method for multiobjective linear-programming problems and its application. IEEE Transactions on Systems, Man, and Cybernetics, SMC-17, 654–661.CrossRefGoogle Scholar
  26. Słowìnski, R. (Ed.) (1998). Fuzzy sets in decision analysis, operations research and statistics. Dordrecht, Boston, London: Kluwer.Google Scholar
  27. Słowìnski, R., & Teghem, J. (Eds.) (1990). Stochastic versus fuzzy approaches to multiobjective mathematical programming under uncertainty. Dordrecht, Boston, London: Kluwer.Google Scholar
  28. Sommer, G., & Pollastschek, M. A. (1978). A fuzzy programming approach to an air pollution regulation problem. In R. Trappl, G. J. Klir, & L. Ricciardi (Eds.), Progress in cybernetics and systems research, (pp. 303–323). John Wiley & Sons, New York.Google Scholar
  29. Verdegay, J. L., & Delgado, M. (Eds.) (1989). The interface between artificial intelligence and operations research in fuzzy environment. Köln: Verlag TÜV Rheinland.Google Scholar
  30. Wierzbicki, A. P. (1980). The use of reference objectives in multiobjective optimization. In G. Fandel, & T. Gal (Eds.), Multiple criteria decision making: Theory and application (pp. 468–486). Berlin: Springer.CrossRefGoogle Scholar
  31. Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8, 338–353.CrossRefGoogle Scholar
  32. Zimmermann, H. -J. (1976). Description and optimization of fuzzy systems. International Journal of General Systems, 2, 209–215.CrossRefGoogle Scholar
  33. Zimmermann, H. -J. (1978). Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems, 1, 45–55.CrossRefGoogle Scholar
  34. Zimmermann, H. -J. (1987). Fuzzy sets, decision-making and expert systems. Boston: Kluwer.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Masatoshi Sakawa
    • 1
  • Hitoshi Yano
    • 2
  • Ichiro Nishizaki
    • 1
  1. 1.Department of System Cybernetics Graduate School of EngineeringHiroshima UniversityHigashi-HiroshimaJapan
  2. 2.Department of Social Sciences Graduate School of Humanities and Social SciencesNagoya City UniversityNagoyaJapan

Personalised recommendations